On defining number of subdivided certain graph by ProQuest

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```									    Scientia Magna
Vol. 6 (2010), No. 1, 110-120

On deﬁning number of subdivided certain graph
†                      ‡
H. Abdollahzadeh Ahangar           and D. A. Mojdeh

† Department of Basic Science, Babol University of Technology, Babol - Iran
‡ Department of Mathematics, University of Tafresh, Tafresh, Iran
E-mail: ha.ahangar@yahoo.com damojdeh@yahoo.com

Abstract In a given graph G = (V, E), a set of vertices S with an assignment of colors to
them is said to be a deﬁning set of the vertex coloring of G if there exists a unique extension
of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A deﬁning set with minimum
cardinality is called a minimum deﬁning set and its cardinality is the deﬁning number. Let
G = Cn+1 1, 4 be circulant graph with V (G) = {v1 , v2 , · · · , vn+1 }. Let G and G be graphs
obtained from G by subdividing of edges vi vi+1 1 ≤ i ≤ n + 1 (mod n + 1) and all of edges of
G respectively. In this note, we study the chromatic and the deﬁning numbers of G, G and
G .
Keywords Chromatic number, circulate graph, subdivided edge, deﬁning number.

§1. Introduction
Throughout this paper, all graphs are ﬁnite, undirected, loopless and without multiple
edges. We refer the reader to [9] for terminology in graph theory. A proper k-coloring of a
graph G is a labeling f : V (G) → {1, 2, · · · , k} such that adjacent vertices have diﬀerent labels.
The labels are colors; the vertices of one color form a color class. The chromatic number of a
graph G, written χ(G), is the least k such that G has a proper k-coloring. A chromatic coloring
of a graph G is a proper coloring of G using χ(G) colors.
In a given graph G = (V, E), a set of vertices S with an assignment of colors to them is
said to be a deﬁning set of the vertex coloring of G if there exists a unique extension of the
colors of S to a c ≥ χ(G) coloring of the vertices of G. A deﬁning set with minimum cardinality
is called a minimum deﬁning set and its cardinality is the deﬁning number denoted by d(G, c).
For more see [3, 5, 6, 7, 8].
The circulate graph Cn+1 1, m is the graph with vertex set {v1 , v2 , · · · , vn+1 } and edge
set {vi vi+j (mod n+1) |i ∈ {1, · · · , n + 1}} and j ∈ {1, m}. It is necessary for circulant graphs
to be connected [2]. Theoretical properties of circulant graphs have been studied extensively
and are surveyed in [1]. The problem of determining chromatic numbers of circulant graphs is
related to periodic colorings of integer distance graphs [4, 5].
Let G = Cn+1 1, 4 be circulant graph with V (G) = {v1 , v2 , · · · , vn+1 }. Let G and G be
graphs obtained from G by subdividing of edges vi vi+1 1 ≤ i ≤ n + 1 (mod (n + 1)) and all of
edges of G respectively.
Vol. 6                         On deﬁning number of subdivided certain graph
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